$q$-变形高斯酉系综：谱矩与亏格型展开

具有 Al-Salam–Carlitz 权重的 $q$-形变酉系综的谱分析

Spectral analysis of $q$-deformed unitary ensembles with the Al-Salam--Carlitz weight

本文研究带 Al-Salam–Carlitz 权重的 $q$-形变随机酉系综，借助 Flajolet–Viennot 理论与匹配组合学得到谱矩正和式及大 $N$ 展开，并给出含两次软边缘相变的极限谱密度。

随机形变高斯酉系综最大特征值的中偏差

Moderate Deviations for the Largest Eigenvalue of a Randomly Deformed Gaussian Unitary Ensemble

该研究针对随机形变高斯酉系综的最大特征值，推导了其在固定耦合 regime 下左右尾的精确对数渐近及相图，建立了一致的Airy近似并结合条件卷积论证完成分析。

q-deformed Character Theory for Infinite-Dimensional Symplectic and Orthogonal Groups

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

Circular Jacobi Ensembles and deformed Verblunsky coefficients

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{δ,β}^{(n)} \prod_{1\leq k<l\leq n}| e^{\iiθ_k}-e^{\iiθ_l}|^β\prod_{j=1}^{n}(1-e^{-\iiθ_j})^δ (1-e^{\iiθ_j})^{\overlineδ} $$ with $\Re δ> -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(α_0, ..., α_{n-1})$. We introduce here a deformation $(γ_0, >..., γ_{n-1})$ of these coeff…

具有离散旋转对称性的指示函数的Toeplitz行列式与随机酉矩阵的幂的渐近展开

Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices

本文针对大N情形下随机酉矩阵m次幂的特征值落在指定弧区间的概率，给出了对应离散旋转对称指示函数Toeplitz行列式的完整渐近展开，解决了此前猜想，并提供了用拓扑递归计算亏格g>0谱曲线含非微扰项的完整渐近展开的实例。

Outliers in the spectrum of large deformed unitarily invariant models

We investigate the asymptotic behavior of the eigenvalues of the sum A+U*BU, where A and B are deterministic N by N Hermitian matrices having respective limiting compactly supported distributions μ, ν, and U is a random N by N unitary matrix distributed according to Haar measure. We assume that A has a fixed number of fixed eigenvalues (spikes) outside the support of μ, whereas the distances between the other eigenvalues of A and the support of μ, and between the eigenvalues of B and the support of ν, uniformly go to zero as N goes to infinity. We establish that only a particular subset of the…